Numerically controlled milling with parabolic profile tools for surface smoothness

ABSTRACT

An arrangement for machining surfaces with a numerically controlled tool in which the tool is provided with a parabolic profile to obtain a substantially smooth surface finish. The workpiece to be machined is moved relative to the cutting tool in substantially a plane of the workpiece. After milling a contour along that plane, the workpiece or the tool is incremented to an adjacent parallel plane for the purpose of milling another contour on the workpiece. The slope of the surface element on the workpiece between the adjacent planes is computed, and the cutting tool is oriented so that the portion of the tool in contact with the workpiece along that surface element, possesses the computed slope. The cutting tool may be formed with cutting edges having straight-line segments interconnected and directed substantially along a parabolic surface. The slopes on the cutting edges of the tool vary in magnitude from substantially zero to infinity.

This is a division of application Ser. No. 801,818 filed May 31, 1977.

BACKGROUND OF THE INVENTION

In the machining of surfaces with conventional cutting tools innumerically controlled milling machines, for example, a substantiallyrough surface finish is obtained after the machining process has beencompleted. This is particularly true when the surfaces being machinedhave varying curvatures.

When using a conventional ball end mill or a flat face end mill, forexample, ridges are formed on the machine surface along the contoursthat are cut with the end mills. These ridges result in applying a roughsurface finish to the machined workpiece.

A basic deficiency of the conventional milling procedures is that theslopes of the workpiece between milled contours is not taken intoaccount. There is no provision in the conventional milling proceduresfor adapting the cutting tool to the varying slopes of the workpiecebetween contours.

Accordingly, it is an object of the present invention to provide anarrangement in which a numerically controlled milling process results ina substantially smooth surface finish of the workpiece.

Another object of the present invention is to provide an arrangement ofthe foregoing character, which may be easily and economically applied.

SUMMARY OF THE INVENTION

The objects of the present invention are achieved by providing a cuttingtool with a substantially parabolic profile. The profile may becomprised of straight-line segments interconnected and directed along aparabolic surface. At the same time, the cutting tool may be formed withcutting edges shaped to result in a smooth parabolic profile of thetool.

To machine a surface of a general shape, a contour is cut on theworkpiece along a predetermined plane. This cutting action results fromrelative movement between the tool and the workpiece. The relativemovement may be obtained by holding the axis of the tool stationarywhile moving the workpiece, or vice versa. Movements of both the tooland the workpiece may also be combined for purposes of obtaining thedesired relative cutting action to generate the contour.

After the contour has been cut, the tool and/or the workpiece isincremented to an adjacent parallel plane for the purpose of cuttinganother contour.

In cutting the contours, in accordance with the present invention, theslopes of the desired machined surface on the workpiece is computed, andthe workpiece is oriented so that the cutting edges on the tool incontact with the workpiece at the contours, possess the computed slope.The computed slope may be selected from a series of slopes available onthe cutting tool with the parabolic profile. The cutting tool serves asa storage or library of cutting slopes in the range of substantiallyzero to infinity. During the milling process, the cutting tool may berotated about its axis so that the cutting edges along the parabolicprofile of the tool apply cutting action to the workpiece.

The novel features which are considered as characteristic for theinvention are set forth in particular to the appended claims. Theinvention itself, however, both as to its construction and its method ofoperation, together with additional objects and advantages thereof, willbe best understood from the following description of specificembodiments when read in connection with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a perspective view and shows a ball end mill in relation to aworkpiece being machined;

FIG. 2 is a perspective view and shows a flat face end mill in relationto a workpiece being machined;

FIG. 3 is a plan view of the surface finish of a workpiece machined witha flat face end mill;

FIG. 4 is a front view of the workpiece of FIG. 3;

FIG. 5 is a side view of the workpiece of FIG. 3;

FIG. 6 is a front view of a workpiece milled with a flat face end millalong contours arranged perpendicular to contours shown on the workpiecein FIG. 3;

FIG. 7 is a plan view of the workpiece of FIG. 6;

FIG. 8 is a side view of the workpiece of FIG. 6;

FIG. 9 is a front view of the surface finish of a workpiece machinedwith a parabolic profile cutting tool, in accordance with the presentinvention;

FIG. 10 is a plan view of the workpiece of FIG. 9;

FIG. 11 is a side view of the workpiece of FIG. 9;

FIG. 12 is a diagrammatic view of a parabolic profile tool, inaccordance with the present invention;

FIG. 13 is another embodiment of the tool of FIG. 12;

FIG. 14 is a graphical representation of slopes applied to the tool ofFIG. 12;

FIG. 15 is a graphical representation of the slopes applied to the toolof FIG. 13;

FIG. 16 is a graphical representation of cutting tools with varyingparabolic profiles;

FIG. 17 is a front view of the surface finish of a workpiece machinedwith a parabolic profile tool, in which the milled contours are normalto the contours of the workpiece shown in FIG. 9;

FIG. 18 is a plan view of the workpiece of FIG. 17; and

FIG. 19 is a side view of the workpiece of FIG. 17.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Referring to FIG. 1, there is shown a block of material 20 which is tobe machined with a cutting tool 22 so as to result in a contouredsurface 24.

If the cutting tool 22 is a ball end mill, having a conventional roundedtip, for example, the surface 24 after machining, will possess ridges 26which result in the surface not being smoothly finished. Thus, aprofiled or contoured surface 24 is formed by first moving the cuttingtool 22 in the x-y plane to form a substantially narrow surface element26 corresponding to the desired shape of the surface 24, at the positionindicated along the z-axis. After having moved in the x-y plane to cutthe surface element 26, the tool 22 is incremented along the z-axis, andthe tool is thereafter moved once again in the x-y plane, for example,for purposes of machining another surface element closely adjacent tothe element 26. By carrying out this process repeatedly of moving thetool in the x-y plane after being incremented or displaced along thez-axis, the desired contoured or profiled surface 24 may be obtained.This method of machining, moreover, makes it possible to obtain asurface of any desired shape.

When the surface 24 is generated by machining with a conventional ballend mill, the surface possesses a series of ridges which mark the pathof the tool and contribute to a resulting roughness of the machinedsurface.

A roughened surface is also obtained when machining with a conventionalflat face end mill, as shown in FIG. 2. When the flat face end mill 28is incremented along the z-axis after having been moved in the x-y planeto generate the surface 30, a stepped finish is obtained on the surfaceof the material 32 being machined. The stepped structure of the roughsurface may be readily seen on a plane 34 passed normal to the machinesurface as may be seen from FIG. 2, FIG. 4 and FIG. 5.

In the plan view of FIG. 3, a small portion 36 of the machine surface 30is shown in enlarged form by the portion 38. As may be seen from theenlarged portion 38 each contour or path of the tool 28 is comprised ofa plurality of straight-lined segments 40, for example. The connectingstraight-lined segments 40 correspond to the short straight-lined pathsof the tool taken to generate a desired contour. Thus, when moving inthe x-y plane, the tool is moved progressively between coordinatesinterconnected by straight lines. By increasing the number of suchcoordinates, and thereby the number of straight lines along the path ofthe tool, the contour may be given a substantially smooth outline in thex-y plane, for example.

A similar roughened surface results when the tool 28 is moved alongcontoured paths in the z-x plane and incremented along the y-axis. Suchan arrangement is shown in FIGS. 6-8, for example. In the view of FIG.6, a small portion 42 of the surface 44 is shown in enlarged form by thesection 46. Straight-lined segments 48 represents again the shortsuccessive paths of the tool 28.

In contrast to the rough surfaces obtained when machining surfaces withconventional cutting tools, the present invention provides for anarrangement in which a substantially smoothly machined surface 50 isobtained, as shown in FIGS. 9-11. In viewing FIGS. 9 and 11, it may beseen that a smooth surface is substantially obtainable by connectingsuccessive planes 52 and 54, for example, by a straight-lined segment56. It may be seen from FIGS. 9 and 11 that the straight-lined segment56 has a particular slope that is different from the slopes of thecorresponding neighboring straight-lined segments interconnectingneighboring planes. Accordingly, if a smooth surface is desired, thecutting tool is required to machine surface elements 56 of varying slopeon the surface to be generated. Thus, the cutting tool must generate onesurface element with a slope 56 which is different from the slope of theneighboring surface element, for example. By varying the slope cut bythe tool, in this manner, a substantially high degree of smoothness maybe obtained over the surface to be machined. Added smoothness may alsobe obtained by subdividing the surface to be machined in a larger numberof contours or planes 52, 54, for example. When subdividing the surfacein a larger number of contours, in this manner, the segments 56 becomecorrespondingly shorter in length.

The tool 58 provided, in accordance with the present invention, formachining contours with different slopes as required by a desired shapeto be generated, is shown in FIG. 12.

The tool 58 is constructed of a surface having a plurality of slopessuccessively interconnected and commencing from a slope of infinity 60to a slope approaching zero at some distance from the end of the tool.The tool 58 is constructed so as to possess straight-lined surfaceelements 62 having slopes corresponding to those that may be requiredfor generating the smooth surface 50 in FIGS. 9-11.

In synthesizing the proper profile to achieve the desired results by thetool shown in FIG. 12, there may be assumed a spacing H between contoursresulting from quantizing the surface to be machined, as may be seen ina plan view of the surface with the contours reproduced thereon. Thetool profile is derived by providing allowable slopes M, such as ∞, 1,1/2, 1/3, etc. if the spacing between contours is designated by doublethe amount or 2H, for example, then the resultant tool profile appearsas shown in FIG. 13. The selection of slopes in the tool profile of FIG.13, corresponds to slopes equal to ∞, 2, 1, 2/3, 1/2, etc. Graphicalvector representations of the slopes on the tool in FIGS. 12 and 13 areshown, respectively, in FIGS. 14 and 15.

If the straight-lined segments or increments on the tool surfaces inFIGS. 12 and 13 are selected sufficiently small, it will be found thatthe profiles of these tools in FIGS. 12 and 13 correspond to parabolicshapes.

To show that the proper tool profile is parabolic, the following proofis constructed in conjunction with FIG. 16. Let the y-ordinates bequantized to Q times the horizontal increments, H. Then as data orcoordinates change horizontally between contours, slopes result equal tothe vertical distance between contours (shown normalized to one) dividedby the allowable horizontal displacement between corresponding points:this may be expressed analytically by, ##EQU1##

From the above relationship, it may be seen that the slope of the N^(th)tool element is Q/N.

To obtain a smoothly machined surface, it is essential to orient thetool so that the tool surface with the proper slope is in contact withthe respective surface element to be machined, so that the desired slopeof the surface element to be machined is carried out by thecorresponding slope on the tool. As a result, each slope on the toolmust be maintained for a distance QH=1 transverse to the tool axis sothat the portion of the cutter at the proper slope occupies preciselythe space between adjacent contours. If the distance along the tool axisover which the slope is held constant is denoted by Δx, ##EQU2## fromwhich it may be seen that the length of the N^(th) element is NH.

The y height at the N^(th) element is N+0.5 since the 0^(th) element ishalf above and half below the tool axis. The total x length at theN^(th) element is the sum of the individual tool portions; i.e.,##EQU3## Thus, the tool shape profile may be expressed as a parameter ofN as shown below. ##EQU4## Solving for N as a function of y

    y(N)=N+1/2

Therefore

    N=y(N)-1/2

Substituting into the expression for x(N).

    ______________________________________                                         ##STR1##                                                                      ##STR2##                                                                                              x(N)       x(N)                                             y(N)    y(N).sup.2                                                                              (computed) (actual)                                  ______________________________________                                                     .5        .25     0        0                                                  1.5       2.25    .25      .25                                   Q = 4        2.5       6.25    .75      .75                                                3.5       12.25   1.5      1.5                                                4.5       20.25   2.5      2.5                                                .5        .25     0        0                                                  1.5       2.25    1        1                                     Q = 1        2.5       6.25    3        3                                                  3.5       12.25   6        6                                                  4.5       20.25   10       10                                    ______________________________________                                    

It may be seen from the above analysis, therefore, that the profile ofthe cutting tool does actually approximate a parabolic shape.Accordingly, the cutting tool need not be profiled with straight-linedsegments as shown in FIG. 16. Instead, the cutting tool may have appliedto it a profile having a substantially parabolic shape. Such a cuttingtool achieves the desired result of providing a smoothly finishedmachined surface in a direction which is transverse to the planes of thecontours. Whereas the roughness of the machine surface can generally bereduced by spacing the contours closer together, when using conventionaltools, for example, the present invention permits wider spacing of suchcontours to obtain satisfactory results. The wider spacing permittedwith the present invention reduces the machining time and thereby thecost of the articles being machined.

One property of the parabolic tool is that the tool diameter at a givendistance back from the tool tip is proportional to the number ofhorizontal quantizing increments between contours. This is illustratedin FIG. 16 by the dashed tool profiles for Q=1, Q=2, Q=4, and Q=8. Atool whose width increases rapidly with increasing distance from the tipas in FIG. 13, is not as suitable for milling fine detail as one with aslower rate of increase corresponding to FIG. 12. Thus, from FIG. 16 itmay be seen that the tool with Q=1 is far superior to the other two forcutting fine detail.

In operation of the present invention, the cutting tool is progressivelypositioned between x-y coordinates along a contour at a given distancealong the z-axis, as shown in FIGS. 9-11. The path of the tool betweenx-y coordinates is computed by means of a conventional computer, forexample, and the instructions or commands for displacements of the toolare applied or stored on tape which is played back at the machine tool.The instructions for guiding the machine tool, in this manner, alsoinclude instructions for selecting the proper slope on the parabolictool and applying the selected slope against the surface to be machinedso as to result in the desired smoothness. After the tool has been movedin the x-y plane to complete the contour, the tool is incremented ordisplaced along the z-axis for the purpose of machining along aneighboring contour with a repetition of the displacements of the toolsimilar to that described for the preceding contour in combination withapplying the proper slope surface on the tool to the surface element orcontour being machined.

It is not essential that the contours lie in x-y planes as illustratedin FIGS. 9-11, to obtain the desired results with the present invention.Equally smooth surfaces may be machined when the contours lie in z-xplanes, as shown in FIGS. 17-19. Thus, a surface 64 may be generated bysubdividing it with contours between planes 66 and 68, for example. Theslope 70 of the contour between the two planes is then machined byplacing the proper surface of the parabolic tool in contact with thecontour being machined. The final result obtained with the arrangementof FIGS. 17-19 will provide substantially the same smoothness asobtained with the arrangement of FIGS. 9-11.

Without further analysis, the foregoing will so fully reveal the gist ofthe present invention that others can, by applying current knowledge,readily adapt it for various applications wihout omitting features that,from the standpoint of prior art, fairly constitute essentialcharacteristics of the generic or specific aspects of this invention,and therefore, such adaptations should and are intended to becomprehended within the meaning and range of equivalence of thefollowing claims.

What is claimed is:
 1. A cutting tool with cutting edges having aparabolic profile, said profile being comprised of cutting surfacesinterconnected to form said parabolic profile, the slopes on saidcutting tool varying in value from zero to infinity, said cutting toolhaving an axis of rotation and a free end, said cutting tool havingcutting edges with slopes varying in value from substantially infinityas said free end of said tool to zero at a predetermined distance alongsaid axis of rotation from the end of said tool, said tool beingselectively positioned for locating the cutting surface conforming tothe desired slope against the surface to be machined.
 2. A cutting toolas defined in claim 1 wherein said cutting edges are comprised of apredetermined number of straight-line segments approximatingsubstantially said parabolic surface.